Algebraic Automata Theory: Homework on Codes and Finite Transducers, DNA Codes - Prof. Nat, Assignments of Mathematics

Download Algebraic Automata Theory: Homework on Codes and Finite Transducers, DNA Codes - Prof. Nat and more Assignments Mathematics in PDF only on Docsity! March 10, 2005 MAD 6616: ALGEBRAIC AUTOMATA THEORY HOMEWORK # 3 DUE MARCH 29TH, 2005 INSTRUCTOR: NATAŠA JONOSKA Choose three problems from each section. I Codes and Finite Transducers. In problems 1-2 determine whether the indicated set is a code, prefix code, suffix code, comma-free code, infix code. 1. L = {ambn|m,n ≥ 1} 2. L = {a4, b, ba2, ab, aba2}. 3. Let ϕ : A∗ → C∗ be an injective morphism. Show that if L ⊆ A∗ is a code, then ϕ(L) is a code over C. And converse, show that if L′ ⊆ C∗ is a code over C, then ϕ−1(L′) is a code over A. If L is a prefix (suffix) code over A, is ϕ(L) a prefix (suffix) code over C? 4. Show that if L ⊆ A∗ is a code then Ln is a code for all integers n > 0. If L is a prefix (suffix), is Ln a prefix (suffix)? 5. Let T be a finite transducer and let D(T ) be the domain of T (see notes #7). Show that D(T ) is a regular set by giving an automaton that recognizes D(T ). 6. Give an algorithm for deciding whether two finite transducers have the same domain. 7. Consider the NSM M′ constructed from the FSA M = (Q, I, T,E) in the proof of Proposition 2.3 (notes # 7). Show that L = L(M) is a code iff M′ is single valued. II DNA codes (The following three exercises are related to the Flower automaton.) 8. (Proposition 1.10, notes 7) Prove that a finite set of words X is a code if and only if the Flower automaton M(X) is unambiguous. 1